The response of a finite beam on a tensionless Pasternak foundation subjected to a harmonic load

The response of an elastic beam on a two-dimensional tensionless Pasternak foundation that is subjected to a central concentrated harmonic load is investigated. The tensionless character of the foundation results in the creation of lift-off regions between the beam and the foundation. Since the contact regions are not known in advance, the problem appears as a nonlinear one, and the calculation of the roots of a nonlinear equation is needed to obtain contact lengths of the beam. The governing equations obtained for the contact and lift-off regions are solved by using the trigonometric-hyperbolic functions. Several numerical examples are provided to show the effects of the foundation stiffness parameters and the frequency parameter on the lift-off regions and the vertical displacements of the beam. It is concluded that more than one contact length may arise in the solution for a fixed frequency parameter, and this parameter may cause to interchange the contact and lift-off regions.